convolution-theorem.py

# Initialize two signals with arbitrary length
n, m = 5, 7
convolution_length = n+m-1

u = np.random.randint(0, 255, n)
v = np.random.randint(0, 255, m)


# Convolve the two signals
cv = sp.signal.convolve(u, v)
assert cv.shape[0] == convolution_length



# Now, apply the Convolution Theorem. 
## Copy and pad signal. Padding is necessary to match the convolution length.
vv = v.copy()
vv.resize(n+m-1, refcheck=False)


uu = u.copy()
uu.resize(n+m-1, refcheck=False)

# Compute FFT
fu = fft(uu)
fv = fft(vv)

# Inverse FFT of frequency signal
fcv = ifft(fu * fv)


# Does the inverse transform of the product of the Fourier transforms of the signals equal the convolution of the two signals?
assert np.allclose(fcv, cv)

this was helpful - thanks!

I tweaked it to support the convolution of any two vectors (of possibly different lengths).

# Initialize a signal
n, m = 5, 7
u = np.random.randint(0, 255, n)
v = np.random.randint(0, 255, m)

# Convolve u and v
cv = sp.signal.convolve(u, v)

assert cv.shape[0] == n+m-1

# Copy and pad signal
vv = v.copy()
vv.resize(n+m-1, refcheck=False)

uu = u.copy()
uu.resize(n+m-1, refcheck=False)

# Compute FFT
fu = fft(uu)
fv = fft(vv)

# Inverse FFT of frequency signal
fcv = ifft(fu * fv)

assert np.allclose(fcv, cv)

@timviera I'm glad it helped you -- thanks for sharing back. I especially enjoyed learning about np.allclose()